1 Introduction

The simulation of high Reynolds number wall-bounded flows is a topic which is attracting intense research within the turbulent shear flow community for a number of reasons: firstly, the ability to establish and test appropriate and proper scaling laws for skin friction for increasing high Reynolds numbers is crucial to determine their asymptotic behaviour, and secondly, the ability to accurately observe and obtain a complete understanding of wall-bounded turbulent flows is impeded by our inability to simulate flows at the high Reynolds numbers that are encountered in most industrial and geo-physical flows [1].

Experiments have fewer limitations on the Reynolds number and can relatively easily capture large-scale information at discrete locations in a flow. However, experimental measurements suffer from low spatial resolution and an inability to resolve the smallest scales that are ultimately responsible for the dissipation of kinetic energy [2,3,4,5]. Additionally, as Reynolds number increases, the difficulty of measuring small scales escalates, particularly in Particle Image Velocimetry (PIV), due to competing requirements between limited spatial domain and spatial resolution for a given PIV image recording array [6]. Surface laser reflections further constrain experimental PIV measurements of wall-bounded turbulence, limiting how close a PIV measurement can be made to the wall [6, 7]. This lack of near wall velocity field is referred to as limited near-wall velocity data, which is an aspect of the quality and/or limitation of the PIV experiment. This was the subject of our preliminary study [8], which was restricted to the direct numerical simulation (DNS) of turbulent channel flow at the low Reynolds number of \(Re_\tau = 180\) and only investigated the effect of limited near-wall velocity data at the inflow of a turbulent channel flow DNS on the spatial development of that turbulent channel flow.

In contrast to experiments, in spatially developing convective turbulent shear flow (TSF) DNSs, the large scales require a large computational domain to allow them to evolve to their correct state, while the small scales adjust relatively quickly to perturbations [9,10,11]. The case of spatially evolving TSF, such as DNS of wall-bounded TSF, also requires a streamwise extent beyond the targeted analysis domain to minimise transients due to recycling used as inflow conditions [12].

Several approaches have been proposed to use measurements as the basis for determining inflow boundary conditions for spatially developing DNS of TSF, falling within the realm of juxtaposing the strengths of simulation and experiments. Druault et al. [13] applied proper orthogonal decomposition (POD) to reconstruct the inflow turbulence from hot-wire measurements with limited spatial resolution and using linear stochastic estimation for the missing velocity data between the measurements. Perret et al. [14] used 3-component-2-dimensional (3C-2D) stereoscopic PIV (SPIV) to reconstruct the inflow turbulence for LES of a turbulent mixing layer. Since the temporal resolution of their PIV was limited, the authors used a low-order dynamic model to represent the temporal evolution of the most energetic modes. The one-point statistics and two-point statistics of the synthesised low-order fields were identical to the measurements and while the anisotropy of the flow and the shear stress were well reproduced using this approach, the long-range correlation levels were over-predicted.

Johansson and Andersson [15] used 3C-2D SPIV to generate the inflow turbulence for turbulent channel flow using the Galerkin projection of Navier–Stokes equation onto the most energetic POD modes of the target flow. They found that low-energy and small-scale modes are necessary to add up to the energetic modes in order to establish the correct levels of dissipation and the correct level of energy distribution within the velocity components.

Data assimilation techniques have emerged as a powerful tool for integrating experimental observations into numerical simulations. Zaki and Wang [16] demonstrated the effectiveness of adjoint-variational techniques in reconstructing flows from limited experimental measurements, such as wall stress and sparse velocity data. These techniques enable accurate reconstruction of flow fields nearly perfectly correlated with ground truth data, even in scenarios where experimental configurations admit multiple flow states or are subject to observation noise. However, assimilating experimental data at higher Reynolds numbers presents persistent challenges, including shorter assimilation windows due to increased Lyapunov exponents and the computational burden of handling a larger volume of observations.

The mentioned studies overlook the incorporation of real or synthetic 3C-2D PIV data to evaluate the effects of low resolution and surface laser reflections, common challenges encountered in PIV, when employed as inflow conditions for DNS. The present study investigates the potential use of time-resolved 3C-2D PIV velocity fields as inflow boundary conditions for a turbulent channel flow DNS. This study is the first to consider the limitations of 3C-2D PIV in its application as a turbulent flow generator for turbulent channel flow DNS at \(Re_\tau = 550\) and 2300. Specifically, it explores the effect of limited near-wall velocity field measurements, limited spatial resolution, and spatially filtered velocity field measurements typical of PIV on the spatial development of turbulent channel flow DNS.

The study employs a unique approach involving two synchronized turbulent channel flow DNS setups. One DNS configuration, termed PCH-DNS, employs periodic boundary conditions in the streamwise direction. The other, termed IOCH-DNS, uses inflow and outflow boundary conditions in the streamwise direction. The difference lies in the fully resolved inflow of the PCH-DNS, which is modified to represent a measured velocity field obtained using 3C-2D PIV. This modification involves spatial filtering, downsampling, and adjustment near the wall to simulate the effect of limited near-wall velocity data. This setup enables a systematic investigation of various PIV qualities as inflow turbulence generators for high-Reynolds-number wall-bounded turbulence DNS.

2 Numerical experiments

DNSs of incompressible turbulent flow of a plane channel were undertaken in this study using two synchronised turbulent channel DNSs–one with periodic inflow/outflow boundary conditions (PCH-DNS), while the other has a specified inflow boundary conditions and a convective outflow condition (IOCH-DNS). For both plane channel flow DNSs, the streamwise, wall-normal and spanwise directions are denoted as x, y and z respectively with the corresponding velocity components denoted as u, v and w, respectively. A fractional-step method [17] is used to solve the incompressible Navier–stokes equations with a staggered, second-order, finite-difference scheme [18]. The equations are advanced in time using a minimal storage third-order Runge–Kutta scheme [19]. The numerical grid for the channel flow DNS has \({N_{x}, N_{y},}\) and \({N}_{z}\) points in the streamwise, wall-normal, and spanwise directions respectively, with the size of the computational domain in the respective directions denoted by \(\textit{L}_{x}\), \(\textit{L}_{y}\), and \(\textit{L}_{z}\). The numerical method and code has been validated in previous turbulent channel flows studies [20,21,22], as well as flat-plate turbulent boundary layer flow investigations [23].

The DNSs for this study were performed at a frictional Reynolds number, \(Re_\tau = \frac{u_\tau \, h}{\nu } = 550\) and 2300, where \(u_\tau = \sqrt{\frac{\tau _w}{\rho }}\) is the friction velocity, with \(\tau _w\) denoting the mean wall shear stress, h is the channel half-width and \(\rho \) the fluid density. The numerical grid characteristics for the two \(Re_\tau \) cases are given in Table 1. The grid spacing in the streamwise (\(\Delta \)x\(^{+}\)) and spanwise directions (\(\Delta \)z\(^{+}\)) is kept constant. The minimum wall-normal grid spacing is located near the wall \(({{\Delta }y}^{+}_{{\hbox {min}}})\) with the maximum wall-normal grid spacing located on the channel centreline \(({{\Delta }y}^{+}_{{\hbox {max}}})\). The grid spacings are given in viscous units, with the viscous length scale given by \({\nu }/u_{\tau }\), where \(\nu \) denotes the kinematic viscosity of the fluid. A constant time-step size denoted by \(\Delta t_{+}=\Delta t\,u^{{\tiny {2}}}_{\tau }/{\nu }\) and given in Table 1 was used.

Table 1 DNS numerical details
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The inflow boundary condition velocity data for the IOCH-DNS is generated from the PCH-DNS at every time step with a convective boundary condition applied at the domain exit for the IOCH-DNS. The velocity statistics for the IOCH-DNS are accumulated after the initial transients are washed out from the computational domain of the IOCH-DNS and it has reached a statistically stationary state. To validate the methodology for the IOCH-DNS the fully resolved inflow data from the PCH-DNS for \(Re_{\tau } = 550\) was provided as in inflow boundary condition to the IOCH-DNS. The first and second-order statistics for the entire IOCH-DNS flow field were compared with those of the PCH-DNS and the results of the previous turbulent channel flow DNS [24]. The results are given in Fig. 1, which show that the mean velocity and fluctuating velocity profiles of IOCH-DNS are identical to those of the PCH-DNS. The pre-multiplied spanwise spectra of the streamwise velocity (\(k_{z}{\phi }_{uu}\)) are presented and compare as shown in Fig. 1c. The energy spectra is averaged in time and the streamwise direction, which shows that the IOCH-DNS is able to reproduce the energy across all the spanwise scales.

Fig. 1
figure 1

Comparison of IOCH-DNS with PCH-DNS at \({Re_{\tau } = 550}\) and the DNS results of [24]: a Mean velocity profile; b Root-mean-squared velocity profiles; c pre-multiplied spanwise spectra of the streamwise velocity, \({k_{z}}{\phi }_{uu}\), as a function of \(y^+\)-the contours are equally spaced between 0.1 to 1.0 of the maximum value of \({k_{z}}{\phi }_{uu}\) of the PCH-DNS; PCH-DNS (black dashed line), IOCH-DNS (colour contour)

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2.1 Methodology to generate limited near-wall inflow data from the PCH-DNS

The near-wall 3C-2D velocity data at the inflow is obtained by filtering out turbulent velocity fluctuations except for those in the zeroth spanwise Fourier mode, which represents the mean velocity. This filtering is applied over a certain distance from the wall up to a specified height above it. This replicates the absence of near-wall data that generally results from the finite PIV interrogation window size and the effect of potential surface laser reflections in a PIV experiment. The threshold height above the plate below where the fluctuations are removed is represented as \({y^{+}_{{th}}}\). In order to investigate the effect of the threshold height above the plate below where the fluctuations are removed, the following typical threshold values are chosen: (i) the top of the viscous sub-layer (\({y^{+}_{{th}} = 5}\)), (ii) within the buffer layer \(({y^{+}_{{th}} = 17})\) and (iii) the beginning of the log-law region \(({y^{+}_{{th}} = 35}).\) Retaining only the mean velocity at the inflow boundary conditions was chosen due to its well-understood and universal nature in the near-wall region. This understanding allows the extension of the mean velocity from PIV to the wall using theoretical and composite profiles [25, 26]. Thus, enabling a focused study on the consequences of losing near-wall fluctuating quantities due to factors such as filtering, low spatial resolution, and surface laser reflections.

The choice of \(y^+\) values in our study aligns with typical ranges. Previous research, as cited in [27, 28], has showcased PIV’s capacity to resolve down to \(y^+ = 0.4\) at \(Re_\tau = 5,400\). This underscores PIV’s effectiveness in measuring below \(y^+ = 35\), extending to \(y^+ = 5\) even at higher Reynolds numbers like \(Re_\tau = 40,000\). Notably, our measurements include data from \(y^+ = 3\), marking one of the highest \(Re_\tau \) wall-bounded spatially resolved near-wall PIV measurements to date [28]. For \(Re_\tau \) values below 40,000, employing PIV with suitable sensors enables measurements below \(y^+ = 1\). These studies collectively underscore the feasibility of wall-bounded PIV measurements at high \(Re_\tau \), reaching into the viscous layer.

The Fourier series representation of the instantaneous velocity components in the spanwise direction is given as:

$$\begin{aligned} u_{i}(t, x, y, z) = \sum _{k_z} \hat{u}_i (t,x,y,k_z) e^{jk_zz}, \end{aligned}$$
(1)

where \( \hat{~} \) denotes the Fourier coefficient, \(k_z\) is the the spanwise wavenumber and the subscript i denoted the coordinate direction of the velocity component such that \(u_1 = u\), \(u_2 = v\) and \(u_3 = w\). Once the velocity fluctuations are removed in all but the zeroth spanwise Fourier mode for \({y^+ \le y^{+}_{{th}}}\), the limited near-wall velocity inflow boundary condition is expressed as:

$$\begin{aligned} u_{i}(t, 0, y, z) = {\left\{ \begin{array}{ll} \hat{u}_i(t, 0, y, 0), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \forall \; \; y^{+} \le y^{+}_{th}, \\ \sum _{k_z} \hat{u}_i (t, 0, y, k_z) e^{jk_zz}, \; \; \forall \; \; y^{+} > y^{+}_{th}. \end{array}\right. } \end{aligned}$$
(2)

2.2 Methodology to generate a 3C-2D PIV velocity field from PCH-DNS for the IOCH-DNS

In order to model the 3C-2D PIV based on the PCH-DNS velocity field, an effective measurement volume is defined for each PIV velocity measurement that corresponds to the PIV interrogation volume. The centroids of these measurement volumes are equally spaced in the wall-normal and spanwise directions, representing the centroids of the PIV interrogation volumes. The interrogation volume has a streamwise length \(\Delta W_x\) and since the velocity plane of interest in the PCH-DNS is the inflow of this DNS which equals the outlet of it, half the streamwise length of the measurement volume \(\Delta W_x/2\) in the streamwise direction extends downstream of the inflow plane, while the other half extends upstream of the outlet plane as illustrated in Fig. 2. Within each measurement volume the PCH-DNS velocity is integrated to yield a filtered velocity at the centroid of each measurement volume. Ultimately, the filtered velocity is required at the computational wall-normal, spanwise grid points of the IOCH-DNS. This is achieved by interpolating the equally spaced and filtered PCH-DNS velocity at the inflow onto the staggered DNS grid using third-order spline interpolation. The trapezoidal rule is used for the numerical integration of the PCH-DNS within the measurement volume. The method is illustrated schematically in Fig. 2.

Mathematically the above description is summarised as follows. Given the DNS velocity field at its discrete grid points:

$$\begin{aligned} u_{i}(x_l, y_m, z_n) \; \; \text {where} \; \; \; l = 0,..,N_x-1; \; m = 0,...,N_y-1; \; n = 0,...,N_z-1 , \end{aligned}$$
(3)

with the equivalent PIV interrogation volume having an effective measurement volume of size \({\Delta }W_{x} {\times } {\Delta }W_{y} {\times } {\Delta }W_{z}\) in the streamwise, wall-normal and spanwise directions, respectively with the centroids of the PIV interrogation volumes equally spaced such that there is 50% PIV interrogation volume overlap, which is typical in PIV analysis, the PIV grid spacing in the wall-normal and spanwise directions is given as

$$\begin{aligned} \Delta y&= \frac{\Delta W_y}{2}, \end{aligned}$$
(4)
$$\begin{aligned} \Delta z&= \frac{\Delta W_z}{2}, \end{aligned}$$
(5)

respectively. Hence, the PIV measurement points containing the filtered velocity data in the wall-normal and spanwise directions are given as

$$\begin{aligned} y_J&= J \Delta y, \quad \text {where} \quad J = 0, \ldots , M-1 \end{aligned}$$
(6)
$$\begin{aligned} z_K&= K \Delta z, \quad \text {where} \quad K = 0, \ldots , N-1. \end{aligned}$$
(7)

respectively. The filtered velocity at these PIV grid points is thus given as

$$\begin{aligned} \begin{aligned} \widetilde{u_j} (0, y_J, z_K )&= \frac{1}{\Delta W_x \times \Delta W_y \times \Delta W_z} \\&\quad \times \left( \int _{0}^{\frac{\Delta W_x}{2}}\int _{y_J - \frac{\Delta W_y}{2}}^{y_J + \frac{\Delta W_y}{2}}\int _{z_K- \frac{\Delta W_z}{2}}^{z_K + \frac{\Delta W_z}{2}} u_i(x, y, z)\, \textbf{d}z\, \textbf{d}y\, \textbf{d}x \right. \\&\quad \left. + \int _{L_x-\frac{\Delta W_x}{2}}^{L_x}\int _{y_J - \frac{\Delta W_y}{2}}^{y_J + \frac{\Delta W_y}{2}}\int _{z_K- \frac{\Delta W_z}{2}}^{z_K + \frac{\Delta W_z}{2}} u_i(x, y, z)\, \textbf{d}z\, \textbf{d}y\, \textbf{d}x\right) , \end{aligned} \end{aligned}$$
(8)

where \(\widetilde{u_i}\) \((0,y_J,z_K)\) is the filtered velocity at the inflow of the IOCH-DNS at the spatial points \((0,y_J,z_K)\). This 3C-2D PIV data is subsequently interpolated onto the DNS wall-normal, spanwise grid using third-order spline interpolation denoted by the function \(f(\widetilde{u_j} )\).

$$\begin{aligned} u_j(0,y_m,z_n) = f(\widetilde{u_j} (0, y_J, z_K )). \end{aligned}$$
(9)

The parameters associated with the filtered inflow velocity are comparable to the previous studies [5, 29] and are given in Table 2 in terms of viscous units.

Fig. 2
figure 2

Schematic diagram illustrating the generation of the filtered velocity inflow boundary condition for IOCH-DNS. Note that f( ) denotes third-order spline interpolation

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The selection of the box filter as the filtering method in the current study stems from its alignment with the inherent discretization process utilized in PIV data analysis. In PIV, the flow field is partitioned into discrete windows, each constituting the smallest unit for flow measurement. Within these windows, the flow is treated as uniformly distributed, allowing for the effective use of smaller windows to enhance resolution.

Moreover, the employment of overlapping windows in PIV to enhance resolution underscores the utility of the box filter approach. While contemporary cross-correlation algorithms may utilize Gaussian functions to achieve sub-pixel resolution, this method approximates the correlation function without altering the underlying spatial averaging characteristic of the windows [30, 31]. Thus, the box filter remains an apt choice, ensuring consistency with the spatial discretization inherent in PIV data processing.

Additionally, it’s worth noting that while traditional PIV operates in two dimensions, our consideration extends to three-dimensional applications, such as tomographic PIV. Even in three-dimensional contexts, the fundamental principle of averaging flow within discrete volumes (or voxels) remains akin to the concept of box filtering, further affirming the suitability of this approach for our study.

Table 2 Parameters of the filtered inflow velocity boundary condition
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3 Results and discussions

3.1 The effect of limited near-wall inflow velocity data on the IOCH-DNS

For the three cases considered where the turbulent velocity fluctuation are removed in all but the zeroth spanwise Fourier mode from the wall up to \({y}^{+}_{{th}}\) = 5, 17 and 35, the statistics of the velocity profiles are generated by averaging in time and the streamwise and spanwise directions for PCH-DNS, whereas, for the IOCH-DNS, statistical velocity profiles are generated by averaging only in time and the spanwise direction due to the in-homogeneity in the streamwise direction in the IOCH-DNS case.

Figure 3 shows the development of the mean-skin friction coefficient for the three cases where the turbulent velocity fluctuations have been removed below \({y}^{+}_{{th}}\) = 5, 17 and 35 for \(Re_\tau = 550\) and below \({y}^{+}_{{th}}\) = 35 for \(Re_\tau = 2300\). These results show that for \(Re_\tau = 550\), \(C_f\) returns to its correct value within \(x \approx 0.1h\) for the \(y^+ = 5\) case, i.e. the turbulence velocity fluctuations have been removed from the viscous sublayer, whereas when the turbulent velocity fluctuation are removed up to approximately half the buffer layer, the correct \(C_f\) is not reached until \(x \approx 2h\). If the turbulent velocity fluctuation up to the start of the log layer are removed the correct value of \(C_f\) is not reached until \(x \approx 6h\).

For the higher \(Re_\tau = 2300\) turbulent channel flow with a much larger log-layer, the removal of the turbulent velocity fluctuations up to \(y^+ = 35\) has a much smaller effect on the streamwise distance need by the IOCH-DNS to recover the correct \(C_f\). As the results in Fig. 3 show, the correct value is reached to within \(~5\%\) by \(x \approx 0.1h\). The larger variability of the mean skin friction found for the \(Re_\tau = 2300\) turbulent channel flow computed using IOCH-DNS is a consequence of the smaller number of statistically independent samples available for this case and recalling that streamwise averaging is not possible since the flow is inhomogeneous in the streamwise developing IOCH-DNS.

Fig. 3
figure 3

Development of mean skin friction coefficient \(C_f\) in IOCH-DNS for \(\textit{y}^{+}_{\hbox {th}}\) = 5, 17 and 35. The \(C_f\) IOCH-DNS results are normalized by the \(C_f\) of the correct value from the PCH-DNS

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Figure 4 shows the wall-normal profiles of the mean velocity and the turbulent fluctuating velocity statistics for \(Re_\tau = 550\) IOCH-DNS, where no turbulence fluctuations are present at the inflow for \(y^+ \le 5\). The use of the inflow velocity without fluctuations up to \(\textit{y}^{+}_{\hbox {th}}\) = 5 has no effects on the downstream mean velocity evolution, while the turbulent fluctuation velocity statistics profiles have fully recovered to their correct distribution by \(x = 0.7 h\).

Fig. 4
figure 4

The statistical velocity profiles of \(Re_\tau = 550\) IOCH-DNS with turbulent velocity fluctuations removed in all but the zeroth mode up to \(\textit{ y}^{+}_{\hbox {th}}\) = 5 at inflow. a Mean velocity profile at x = 0.7h, b Root-mean-squared fluctuating velocity profiles-same colour coding as a

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Fig. 5
figure 5

The statistical velocity profiles of \(Re_\tau = 550\) IOCH-DNS with turbulent velocity fluctuations removed in all but the zeroth mode up to \(\textit{ y}^{+}_{\hbox {th}}\) = 17 at inflow. a Mean velocity profile at x = 0.7h and x = 1.5h, b Root-mean-squared fluctuating velocity profiles-same colour coding as a

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Fig. 6
figure 6

The statistical velocity profiles of \(Re_\tau = 550\) IOCH-DNS with turbulent velocity fluctuations removed in all but the zeroth mode up to \(\textit{ y}^{+}_{\hbox {th}}\) = 35 at inflow. a Mean velocity profile at x = 1.5h and x = 3 h, b Root-mean-squared fluctuating velocity profiles-same colour coding as a, c Mean velocity profile at x = 6 h and x = 12 h, d Root-mean-squared fluctuating velocity profiles-same colour coding as c

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For the case where no turbulence fluctuations are present at the inflow for \(y^+ \le 17\) for \(Re_\tau = 550\), the turbulent velocity fluctuations statistics profiles require additional downstream evolution compared to the \(\textit{y}^{+}_{\hbox {th}}\)= 5 case to reach their correct wall-normal distributions as shown in Fig. 5. This case shows that the near-wall cycle has been disrupted at the inflow to the IOCH-DNS and the mean velocity and fluctuating velocity profiles are only recovered at x = 1.5h.

Fig. 7
figure 7

The pre-multiplied spanwise spectra of the streamwise velocity, \({k_{z}}{\phi }_{uu}\) for \(Re_\tau = 550\), as a function of y at various streamwise locations, with energy removed in all but the zeroth mode up to \(\textit{ y}^{+}_{\hbox {th}}\) at inflow. The contours are 0.1 to 1.0 of the maximum value of \({k_{z}}{\phi }_{uu}\) of PCH-DNS, PCH-DNS (black dashed line), IOCH-DNS (contour). a \(\textit{ y}^{+}_{\hbox {th}}\) = 5, b \(\textit{y}^{+}_{\hbox {th}}\) = 17, and c \(\textit{y}^{+}_{\hbox {th}}\) = 35

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Fig. 8
figure 8

Development of the maximum of streamwise premultiplied spectra \(k_z\phi _{uu}\) in IOCH-DNS for \(y^{+}_{th} = 17, 35\). The spectra are normalized by their respective maximum values obtained from the PCH-DNS

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When the turbulent velocity fluctuations are removed at the inflow of the IOCH-DNS for \(y^+ \le 35\), the mean velocity profile in the inner region has not recovered by x = 3 h as shown in Fig. 6a, and while the turbulent fluctuating velocity statistics have nearly developed at this downstream location as shown in Fig. 6b, the streamwise turbulent fluctuations are more intense than the correct corresponding ones of the PCH-DNS. In contrast, the wall-normal and spanwise turbulent fluctuations are less intense than the correct corresponding ones of PCH-DNS at x = 1.5h, but are consistent with the correct PCH-DNS distributions by x = 3 h. Figure 6c shows the mean velocity profiles at x = 0, 6 h, 12 h with the mean velocity profile only recovering to its correct state by x = 6 h. Here, the streamwise turbulent velocity fluctuations are still more intense than the correct PCH-DNS values. As the results in Fig. 6d show, this is still found to be the case even at x = 12 h.

Figure 7 presents the pre-multiplied spanwise energy spectra of the streamwise velocity \({k_{z}}{\phi }_{uu}\) for IOCH-DNS with limited near-wall inflow velocity data. The correct spectra are the result of averaging in time and the streamwise direction for PCH-DNS, whereas, they are only the result of temporal averaging for IOCH-DNS. The flow scales in the viscous sub-layer (case: \(\textit{y}^{+}_{\hbox {th}}\) \(=\) 5) are removed as shown in Fig. 7a. As the channel flow evolves downstream from the inflow, it is observed that at x = 0.5h the turbulent flow scales have recovered to their correct values. This suggests that half a channel width of downstream development (x = 0.5h) is sufficient to recover the turbulent velocity fluctuations to their correct value at the appropriate scale when the turbulent velocity fluctuations have been removed from the viscous sub-layer.

However, when the turbulence velocity fluctuations are removed up to the buffer layer, i.e. case: \(\textit{y}^{+}_{\hbox {th}}\) \(=\) 17, and although the flow evolves to recover the disrupted near-wall mechanism as shown in Fig. 7b, the near-wall velocity scales have not recovered completely by x = 0.7h. They only fully recover by x = 1.5h where the IOCH-DNS is able to maintain the turbulent velocity scales and energy levels similar to the correct PCH-DNS results.

When the turbulent velocity fluctuations are removed up to \(y^+ = 35\), much of the near-wall cycle is removed [32,33,34,35]. As the turbulent channel flow develops downstream, it takes a longer streamwise domain for the turbulent channel flow to recover the near-wall cycle. At x = 0.7h, the outer region remains the same as at the inflow, while the near-wall turbulent scales develop. At x = 1.5h the near-wall peak is recovered, however, this peak is higher than the correct one of the PCH-DNS and the small scales are not yet completely recovered. This is consistent with the strengthened streamwise turbulent velocity fluctuations and attenuated spanwise and wall-normal turbulent velocity fluctuations as shown in Fig. 6. The turbulent near-wall flow scales have nearly recovered by x = 6 h, however, similarly to the streamwise velocity fluctuations, the peak of the \({k_{z}}{\phi }_{uu}\) spectra is still higher than the corresponding correct one from the PCH-DNS.

The maximum of premultiplied spectra of streamwise velocity is shown in Fig. 8 and it aligns with the above observations from the \(k_z\phi _{uu}\) spectra. When near-wall fluctuations are disrupted at the inflow (\(y^+_{th} = 17\)), the flow requires approximately \(x = 1.5h\) to reach the spectral maximum akin to the base flow. However, complete removal of the near-wall cycle necessitates a longer domain, with the flow taking more than \(x = 2\,h\) to represent the spectral maximum similar to that of the base flow. It is to be noted that the spectral maximum of the \(y^+_{th} = 5\) case is excluded from Fig. 8 as the spectral peak remains unchanged at the inflow when fluctuations are removed at \(y^+_{th} = 5\), as seen in Fig. 7a. Moreover, the variability in the spectral maximum for the IOCH-DNS case is influenced by the limited availability of statistically independent samples and the inability to perform streamwise averaging due to the flow’s inhomogeneity in the streamwise developing IOCH-DNS.

3.2 The effect of limited spatial resolution of the inflow 3C-2D PIV data on the IOCH-DNS

In experiments with physical (e.g. hot-wires) or optical (e.g. LDA, PIV) probes, the size of the measurement volume is finite. Hence, turbulent velocity fluctuations with length scales smaller than the measurement volume are filtered out during the measurement. To capture the impact of this filtering at every time step the PCH-DNS velocity field is integrated over the appropriate measurement volume and interpolated onto the numerical wall-normal spanwise staggered grid points at the inflow boundary condition of the IOCH-DNS as described in Sect. 2.2.

Figure 9 shows the effect of the spatially filtered velocity inflow data due to the different measurement volumes (as detailed in Table 2) on the development of the mean skin friction coefficient, \(C_f\), for \(Re_\tau = 550\) and one for the case at \(Re_\tau = 2200\).

These results show that for \(Re_\tau = 550\), the \(C_f\) returns to its correct value within \(x \approx 6h\) for both the R550-IW16 and R550-IW32 cases. However, the initially more filtered inflow velocity field of the R550-IW32 case results in an initially larger under estimated \(C_f\) at around \(x \approx 0.5h\) compared to the R550-IW16 case. In contrast, the higher \(Re_\tau = 2300\) IOCH-DNS with the velocity field filtered to the same level in viscous units as the \(Re_\tau = 550\) R550-IW32 case results in the correct \(C_f\) value being reached within \(x \approx 1.5h\).

Figure 10 shows the mean velocity and statistics of the fluctuating velocity profiles for the R550-IW16 case. Figure 10b clearly shows that the near-wall turbulent velocity fluctuations are not resolved at the inflow plane (i.e. x = 0 h) due to spatial filtering. Downstream of the inflow, at x = 1.5h, the mean velocity profile is not completely recovered in the inner-region as demonstrated by the results presented in Fig. 10a. Figure 10b shows that the profiles of the statistics of the turbulent fluctuating velocity are developing towards their correct distribution at x = 0.7h, but are still lower in the near-wall region.

At x = 1.5h, the statistics of the streamwise turbulent velocity fluctuations is higher than their corresponding correct PCH-DNS values, while the statistics of the spanwise and wall-normal turbulent velocity fluctuations are lower than their corresponding correct PCH-DNS values. At x = 6 h, the profiles of the mean and the statistics of the turbulent velocity fluctuations are fully recovered to their corresponding correct values as demonstrated in Fig. 10c, d, except for the statistics of the spanwise turbulent velocity fluctuations in the near-wall region, which are higher than their corresponding correct PCH-DNS values. Close to the exit of the DNS domain at (x = 12 h), all profiles of the statistics of the turbulent velocity fluctuations are coincident with their corresponding correct PCH-DNS ones and are therefore, completely and correctly recovered.

Fig. 9
figure 9

Development of the mean skin friction coefficient \(C_f\) in IOCH-DNS for the different spatially filtered inflow conditions. The \(C_{f}\) IOCH-DNS results are normalized by the correct \(C_{f}\) value from the PCH-DNS

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Fig. 10
figure 10

The statistical velocity profiles of R550-IW16 IOCH-DNS: a Mean velocity profile at x = 0.7h and x = 1.5h; b Root-mean-squared fluctuating velocity profiles at x = 0.7h and x = 1.5h; c Mean velocity profile at x = 6 h and x = 12 h; d Root-mean-squared fluctuating velocity profiles at x = 6 h and x = 12 h

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The effect of a larger interrogation window has also been investigated for \(Re_\tau = 550\), i.e. the R550-IW32 case. The profiles of the turbulent fluctuating velocity statistics are presented in Fig. 11. Figure 11b shows that the near-wall turbulent velocity fluctuations are further attenuated at the inflow (x = 0 h) compared to the R550-IW16 case due to the larger interrogation window and thus, the additional filtering. As the flow develops downstream, at x = 1.5h, the mean velocity profile is not completely recovered in the inner-region as demonstrated by the results presented in Fig. 11a.

At x = 1.5h the profiles of the statistics of the turbulent velocity fluctuations are developing towards their correct distributions. However, it is found that the turbulent velocity fluctuations are lower in the near-wall region as shown in Fig. 11b with this effect being larger for the spanwise turbulent velocity fluctuation. At x = 6 h, the profiles of the mean and statistics of the turbulent velocity fluctuations are recovered as shown in Fig. 11c, d. Yet, similarly to the R550-IW16 case, the spanwise turbulent velocity fluctuations in the near-wall region are higher than the corresponding correct PCH-DNS ones. The profiles of the statistics of the spanwise and wall-normal turbulent velocity fluctuations collapse to their corresponding correct PCH-DNS ones and are completely recovered close to the exit of the DNS domain at (x = 12 h) with the exception of the turbulent streamwise velocity fluctuations, which are higher than their corresponding PCH-DNS values. It is worth noting that at higher Reynolds numbers, the separation of scales between the near-wall and outer layer turbulence becomes significant and the measurement of small scales becomes complicated because of the limited spatial resolution possible in the experiments.

The velocity statistics profiles of the R2300-IW32 case at the higher Reynolds number of \({Re_{\tau } = 2300}\) are shown in Fig. 12. The filtered near-wall turbulent velocity fluctuations at the inflow, i.e. (x = 0 h), evolve quicker for R2300-IW32 case than the R550-IW32 case as shown in (Figs. 11 and 12). At x = 0.4h, the wall-normal and spanwise turbulent velocity fluctuations are recovering, but with lower values. By x = 3 h, the near-wall turbulent streamwise velocity fluctuations have recovered, whereas the outer layer fluctuation have not yet and are lower than the correct PCH-DNS values. The observed oscillations in the outer layer of the statistics obervable in Fig. 12 are caused by a lack of IOCH-DNS samples and hence, a higher uncertainty of the statistics to the same level of uncertainty as the lower Reynolds number cases.

Fig. 11
figure 11

The statistical velocity profiles of R550-IW32 IOCH-DNS: a Mean velocity profile at x = 0.7h and x = 1.5h; b Root-mean-squared fluctuating velocity profiles at x = 0.7h and x = 1.5h; c Mean velocity profile at x = 6 h and x = 12 h; d Root-mean-squared fluctuating velocity profiles at x = 6 h and x = 12 h

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Fig. 12
figure 12

The statistical velocity profiles of R2300-IW32 IOCH-DNS: a Mean velocity profile at x = 0.4h and x = 0.7h; b Root-mean-squared fluctuating velocity profiles at x = 0.4h and x = 0.7h; c Mean velocity profile at x = 3 h and x = 12 h; d Root-mean-squared fluctuating velocity profiles at x = 3 h and x = 12 h

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Fig. 13
figure 13

The pre-multiplied spanwise spectra of the streamwise velocity, \({k_{z}}{\phi }_{uu}\), as a function of y at various streamwise locations. The contours are 0.1 to 1.0 of the maximum value of \({k_{z}}{\phi }_{uu}\) of PCH-DNS, PCH-DNS (black dashed line), IOCH-DNS (contour): a R550-IW16; b R550-IW32

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Fig. 14
figure 14

The pre-multiplied spanwise spectra of the wall-normal velocity, \({k_{z}}{\phi }_{vv}\), as a function of y at various streamwise locations. The contours are 0.1 to 1.0 of the maximum value of \({k_{z}}{\phi }_{vv}\) of PCH-DNS, PCH-DNS (black dashed line), IOCH-DNS (contour): a R550-IW16; b R550-IW32

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Fig. 15
figure 15

The pre-multiplied spanwise spectra of the spanwise velocity, \({k_{z}}{\phi }_{ww}\), as a function of y at various streamwise locations. The contours are 0.1 to 1.0 of the maximum value of \({k_{z}}{\phi }_{ww}\) of PCH-DNS, PCH-DNS (black dashed line), IOCH-DNS (contour): a R550-IW16; b R550-IW32

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Figure 13a presents the pre-multiplied spanwise energy spectra of the streamwise turbulent velocity fluctuations (\({k_{z}}{\phi }_{uu}\)) for the R550-IW16 case. The filtered inflow result at (x = 0 h) shows that small scales with a wavelength less than \({\lambda }_{z}/h {\ \approx } 0.12\) are not resolved, which is consistent with the attenuation of \({{u}^{+}_{rms}}\) shown in Fig. 10a. As the flow evolves downstream from the inflow, at x= 0.7h, the peak of the energy spectra is recovered. However, near-wall flow scales (y\(^{+}\) \(\ \le \) 15) with a wavelength less than \(\lambda _{z}/h {\ \approx } 0.09\) are not recovered. Further downstream at x = 1.5h, the near-wall flow scales (y\(^{+}\) \(\ \le \) 10) with a wavelength less than \(\lambda _{z}/h {\ \approx } 0.08\) are still not completely recovered. The flow scales are fully recovered at \(x = 3h\), along with the energy at these resolved scales. Close to the outflow of the DNS domain at x = 12 h, the recovered all the flow scales have recovered their correct energy.

Fig. 16
figure 16

Development of the maximum of streamwise premultiplied spectra \(k_z\phi _{uu}\) in IOCH-DNS for the different spatially filtered inflow conditions. The spectra are normalized by their respective maximum values obtained from the PCH-DNS

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Figure 13b shows the pre-multiplied spanwise energy spectra for the R550-IW32 case. The small scales with a wavelength less than \(\lambda _{z}/h {\ \approx } 0.2\) are not resolved at the inflow, i.e. x = 0 h. At x = 0.7h, the near-wall flow scales (y\(^{+}\) \(\ \le \) 15) with a wavelength less than \(\lambda _{z}/h {\ \approx } 0.12\) are not completely recovered, which are higher than those of the R550-IW16 case. This shows that as the spatial resolution 3C-2D PIV at the inflow decreases the flow takes a longer downstream development to evolve the small scales correctly. As the flow evolves downstream, the peak of the energy spectra also increases and is higher than the corresponding correct PCH-DNS peak at x = 1.5h and 3 h. The flow scales are fully recovered at x = 6 h. However, the peak of the spectra remains higher than the corresponding correct one of the PCH-DNS. The evolution of the pre-multiplied spanwise energy spectra of the wall-normal velocity fluctuations (\({k_{z}}{\phi }_{vv}\)) is shown in Fig. 14. Unlike the streamwise velocity spectra, where the near-wall peak is close to the wall (y\(^{+}\) \(\ \approx \) 15), the peaks in the wall-normal and spanwise velocity spectra are away from the wall (y\(^{+}\) \(\ \approx 50 {-} 70\)). Since the peak of the \({k_{z}}{\phi }_{vv}\) energy spectra is further away from the wall compared to \({k_{z}}{\phi }_{uu}\), the filtered inflow maintains the higher energy levels equivalent to the correct PCH-DNS values, and only the small scales are filtered out. The filtered scales continue recovering with downstream development, and by x = 3 h the small scales have recovered completely to the level of the R550-IW16 case. However, the energy levels are still not the corresponding correct values of the PCH-DNS. While for the R550-IW32 case, the small scales have recovered qualitatively by x = 6 h, the peak of the spectra is still higher than the corresponding correct PCH-DNS values.

Similarly, in the evolution of the pre-multiplied energy spectra of the spanwise turbulent velocity fluctuations (\({k_{z}}{\phi }_{ww}\)), it is observed that the filtered small scales are qualitatively recovered by x = 3 h and x = 6 h for R550-IW16 and R550-IW32, respectively. However, the spanwise energy levels are not the corresponding correct values of the PCH-DNS as demonstrated by the results shown in Fig. 15.

Figure 16 illustrates the maximum of the premultiplied spectra of streamwise velocity at the location of the maximum for the periodic channel flow. It is evident from Fig. 16 that it aligns with the findings from the premultiplied \(\phi _{uu}\) spectra of the R550-IW16 and R550-IW32 cases presented in Fig. 13a, b, respectively. Specifically, the R550-IW16 case reaches the spectral maximum of the base flow PCH-DNS faster than the R550-IW32 case. This suggests that as the spatial resolution at the inlet decreases, the flow may require a longer streamwise domain to reach a fully developed state, as indicated by the slower recovery of the spectral maximum in the R550-IW32 case compared to the R550-IW16 case. Additionally, it’s worth noting that the larger variability observed in the spectral maximum for the IOCH-DNS case stems from the limited availability of statistically independent samples. Unlike the periodic channel flow, streamwise averaging is not feasible in the inhomogeneous streamwise developing IOCH-DNS setup, contributing to this variability.

Fig. 17
figure 17

The pre-multiplied spanwise spectra of the streamwise velocity, \({k_{z}}{\phi }_{uu}\), as a function of y at various streamwise locations for R2300-IW32 case. The contours are 0.2, 0.4, 0.6, 0.8 and 1.0 of the maximum value of \({k_{z}}{\phi }_{uu}\) of PCH-DNS, PCH-DNS (black dashed line), IOCH-DNS (colour lines)

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Figure 17 presents the pre-multiplied spanwise energy spectra of the streamwise velocity fluctuations (\({k_{z}}{\phi }_{uu}\)) for the R2300-IW32 case. The small scales with \({\lambda }_{z}/h {\ \le } 0.05\) are not resolved at the inflow x = 0 h. However, the small scales that were removed, recovered earlier for the R2300-IW32 case compared to R550-IW32 case. This suggests that for given filtering of the smaller scales, a greater range of scale at higher Re means that a greater fraction of the overall turbulent kinetic energy is retained at the inflow, which thus results in faster recovery. At \(x = 0.7h\), the flow scales are qualitatively fully recovered, and the energy at the resolved scales is also fully recovered.

4 Concluding remarks

This paper reports a systematic study investigating how experimental data in the form of a time-resolved 3C-2D velocity plane in the streamwise direction of a spatially developing wall-bounded flow can be employed to generate the inflow boundary condition for wall-bounded flow DNS.

Specifically, this study has been undertaken using turbulent channel flow DNS and has investigated the effect of limited near-wall inflow velocity data and the effect of the limited spatial resolution of the time-resolved 3C-2D PIV inflow velocity data on the streamwise development of turbulent velocity statistics and pre-multiplied spanwise velocity energy spectra.

When the near-wall cycle is completely removed at the inflow, i.e. inflow with all turbulent velocity fluctuations removed up to \(y^+ = 35\), the streamwise fluctuations are energised and the spanwise velocity fluctuations are weakened in the process of recovery of the turbulent fluctuations. As the flow approaches the correct PCH-DNS state, i.e. at x \(\approx \) 6 h, the wall-normal and spanwise velocity fluctuations are recovered, however, the streamwise fluctuations are amplified. Similarly, the spectral peak of the streamwise velocity spectra is higher than the corresponding correct PCH-DNS ones.

When the small scales \(({\lambda }_{z}/h \ \le 0.2)\) are filtered out at the inflow, the mean velocity profile is not completely recovered in the inner-region at x = 1.5h and the streamwise fluctuations are higher than the PCH-DNS, while the spanwise and wall-normal fluctuations are weakened. At x = 6 h, the mean-velocity and streamwise fluctuating velocity profiles are recovered, but the spanwise fluctuations are higher than the corresponding correct PCH-DNS ones in the near-wall region. The spanwise energy spectra shows that the spectral peak increase as the flow evolves downstream, and is higher than the correct PCH-DNS one, even when the small scales are completely recovered. This effect is larger in the spanwise energy spectra of spanwise velocity (\({k_{z}}{\phi }_{ww}\)) where the spectral peak is higher than the corresponding correct PCH-DNS one for x \(\ge \) 3 h. The downstream development of the spanwise energy spectra indicates that the recovery of small scales requires a longer domain as the spatial resolution of the inflow 3C-2D PIV decreases.

The PIV interrogation volumes specified in Table 2 for the the R2300-IW32 case are nowadays possible with the available sensor technology. This opens up the possibility of using time-resolved 3C-2D PIV measurements as a realistic inflow to a spatially developing direct numerical simulation of high Reynolds number wall-bounded flow, not just channel flow, but also turbulent boundary layer (TBL) flow developing within different pressure gradient environments. The results of this study show that the mean wall shear stress will be predicted correctly within 1.5h for a channel flow and by extension to within \(1.5\delta \) for a TBL under these experimental measurement constraints. However, the correct turbulent velocity at all scales will take a significantly longer downstream development, conservatively up to at least \(x \approx 12\,h\) for a turbulent channel flow and similarly by extension up to \(x \approx 12 \delta \) for a TBL.

At first observation, this might appear as an extensive use of computational resources, however, given that many DNSs of TBL flows extend to 30–40 initial \(\delta \) and beyond, and furthermore, that there is a significant cost of getting to an initial state of a high Reynolds number TBL, if at all possible, the approach proposed and investigated here of using experimental 3C-2D PIV inflow data and losing the first 25% of the domain is most likely a more efficient approach to high Reynolds number TBL DNS, particularly, if the rate of recovery continues to increase with Reynolds number, as well as, (1) opening up the potential to study the effect of realistic inflows with all their disturbances and (2) enhancing experimental measurements via this data assimilation approach, by yielding additional data such as the pressure field and correctly resolved turbulent flow fields, which are all consequently available from the DNS of the spatially developing wall-bounded flow.